Well, I'm not sure what its asking for exactly but the tangential component of the e-field in a conductor must be zero. If not, the electrons in the conductor rearrange themselves to make it zero. The fact that its zero everywhere means that it is continuous obviously. But perhaps they want a proof of this fact in some way which proves continuity without showing that its zero everywhere?

The continuity of tangential component of an electrostatic field is true for the boundary of any two materials. In a special case when one side of the boundary is conductor, on both sides, the tangential component is zero ( and of course still continuous ).